If are such that then for all . Now suppose there are scalars such that where then we get where then , and so ... . Which means they're l.i. and therefore a basis.
The question asks:
Prove that if (v1, ... , vn) is a basis of V, then so is(v1, v2-v1, ... , vn - vn-1).
I made the initial assumption that if V remained the same number that by subtracting v1, youd always end up with v1, and so on. But obviously I came to the conclusion that this would have to be wrong, because the set of (v1, .. , vn) could represent any numbers. So I have no idea how to tackle this one, any help would be greatly appreciated. Thanks.
Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?
You are given that the first n vectors form a basis for V so you know that V has dimension n. You are given another n vectors in V. To show that they are a basis for V you need only show that they span V or that they are independent.It would almost appear as if that is proving that they are a basis rather than specifically a basis for V.